Volume Thresholds for Gaussian and Spherical Random Polytopes and Their Duals

Let g be a Gaussian random vector in R. Let N = N(n) be a positive integer and let KN be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes VN := E vol(KN ∩RB 2 )/ vol(RB 2 ). For a large range of R = R(n), we establish a sharp threshold for N , above which VN → 1 as n → ∞, and below which VN → 0 as n → ∞. We also consider the case when KN is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0, 1) and R = 1. Lastly, we prove complementary results for polytopes generated by random facets.